Question

The equation of circle whose diameter is the common chord of two circles x²+y²+2ax and x²+y²+2by

Answer 1

Answer:

(b²+a²)(x²+y²) +2ab(bx+ay)=0.

Step-by-step explanation:

Let the circles be S₁ = x²+y²+2ax=0 and

S₂ = x²+y²+2by=0

Then the common chord of the 2 crcles is given by S₁-S₂=0

i.e., ax-by =0 let it be L

Now,

Equation of any circle passing through points of intersection of a given circle S and line L is given by S+λL=0 where L is constant .

So, required equation of circle is  x²+y²+2ax + λ(ax-by) = 0

(here , we can consider circle S₂ as well, still we will end up with same)

x²+y² + x(2a+λa) -λby = 0

So, center is (-(2a+λa)/2, λb/2)

But Since L is diameter center should lie on L

Thus, -a(2a+λa)/2 = bλb/2

=>2a² + λa² = -λb²

=>λ(b²+a²)=-2a²

=>λ=-2a²/(b²+a²)

x²+y² + 2ab²/(b²+a²)x +2a²b/(b²a²)y = 0

(b²+a²)(x²+y²) +2ab(bx+ay)=0...Ans